This is a good question.
I'm not an anesthesiologist so can't comment on 1/2 of your question.
Personally, I find the stress low-medium. We are not saving lives, but rather quality of life. Nobody is dying on us unless we have a rare complication during a procedure or a medication overdose.
We deal with a lot of patients with personality disorders, behavioral problems, addictive personalities. They can cause a lot of headaches.
Some of us do inpatient pain medicine as well, which can vary in stress quite a bit depending on your inpatient population (cancer vs. trauma vs. chronic pain patients).
My anesthesiologist buddies sometimes find clinical pain medicine more stressful than general. They tell me general has pretty consistently good outcomes nowadays. The outcomes in chronic pain are anything but consistently good....